Investigation of Piezoelectric Ringing Frequency Response of Beta Barium Borate Crystals

crystals

Article Investigation of Piezoelectric Ringing Frequency Response of Beta Barium Borate Crystals

Giedrius Sinkevicius 1,2,* and Algirdas Baskys 2

1 Department of Material Science and Electrical Engineering, Center for Physical Sciences and Technology, Sauletekio˙ av. 3, LT- 10257 Vilnius, Lithuania 2 Department of Computer Science and Communications Technologies, Vilnius Gediminas Technical University, Naugarduko st. 41, LT- 03227 Vilnius, Lithuania; [email protected] * Correspondence: [email protected]; Tel.: +370-620-81193

Received: 19 December 2018; Accepted: 15 January 2019; Published: 17 January 2019

Abstract: The piezoelectric ringing phenomenon in Pockels cells based on the beta barium borate crystals was analyzed in this work. The investigation results show that piezoelectric ringing is caused by multiple high voltage pulses with a frequency in the range from 10 kHz up to 1 MHz. Experimental investigation of frequency response and Discrete Fourier transformation was used for analysis. The method of piezoelectric ringing investigation based on the analysis of difference of real and simulated optical signals spectrums was proposed. The investigations were performed for crystals with 3 × 3 × 25 mm, 4 × 4 × 25 mm and 4 × 4 × 20 mm dimensions. It was estimated that piezoelectric ringing in the beta barium borate crystal with dimensions of 3 × 3 mm × 25 mm occurred at the 150, 205, 445, 600 and 750 kHz frequencies of high voltage pulses.

Keywords: Pockels cell; piezo-electric ringing; beta barium borate

1. Introduction A wide variety of optical modulators are used in laser applications [1–7]. However, only modulators based on electro-optic and acousto-optic principles are used for high power laser system applications. The electro-optic modulation principle is based on the Pockels effect, which presents the change of refractive index in non-centrosymmetric crystals under the influence of an external electric field. The change in refractive index induces a polarization change of a beam that travels though the crystal of the Pockels cell. This feature allows us to use the Pockels cell as a voltage controlled half-wave plate [8]. Acousto-optic modulation principle is based on Debye-Sears effect combined with Bragg configuration. The mechanical oscillation of piezoelectric wafer on the side of the crystal creates pressure, which increase the refraction, diffraction and interference inside the crystal [8,9]. This effect allows us to employ the acousto-optic modulator as a beam deflector. The property of any optical modulator is the ability to let through or shut out the laser beam that travels through the modulator. The measure of this property for the electro-optic modulators is contrast ratio; for the acousto-optic modulators - diffraction efficiency. Electro-optical modulators contrast ratio varies from 1000:1 to 2000:1 [10–12] and acousto-optic modulators diffraction efficiency from 30 % to 80 % [4,13–15]. An essential parameter of optical modulators is the rise and the fall time of the modulator. The rise and fall time of electro-optical modulators can reach 10 ns [4,12] and it varies from 200 ns down to 56 ns for acousto-optic modulators [14,15]. It should be noted that the rise and fall time of the acousto-optic modulator depends on the sound velocity in the crystal and the waist diameter of the beam [14]. The rise and fall time of an electro-optical modulators based on the Pockels cell is defined by the rise

Crystals 2019, 9, 49; doi:10.3390/cryst9010049 www.mdpi.com/journal/crystals Crystals 2019, 9, 49 2 of 10 and fall time of the high voltage pulse that is applied to the modulator. The required amplitude of the pulse for operation of any type of Pockels cell depends on the material, dimensions of the crystal and wavelength of the laser beam that travels through the Pockels cell crystal [16,17]. There are two types of Pockels cells: transverse and longitudinal [8]. For longitudinal Pockels cells, half-wave voltage (amplitude of pulse, at which the laser beam polarization has been rotated by 90◦) may vary from 4kV to 10 kV [18]. For transverse Pockels cells, half-wave voltage varies between 1 kV and 8 kV [19,20]. Pockels cell rise/fall time depends on two factors: capacitance of the crystal and applied voltage rise/fall time. In order to achieve a shorter duration of rise/fall time of the Pockels cell, it is necessary to decrease the values of these parameters. The capacitance is determined by the dimensions and material of the crystal; therefore, the main way for the reduction of the rise/fall time of Pockels cell is improvement of the electronics of the high voltage driver. There are only a few usable concepts for high voltage driver design: metal-oxide-semiconductor field-effect transistors (MOSFET) or bipolar avalanche transistors connected in series [21]. The high voltage drivers based on the MOSFETs connected in series provide the laser beam modulation frequency up to 1 MHz with the 30 ns rise/fall time of pulse and adjustable pulse width [22–27]. The bipolar avalanche high voltage drivers allow us to achieve rise time from 7 ns down to 240 ps [28–35]. However, the fall time of the pulse generated by the bipolar avalanche high voltage drivers may be 10 times longer than the rise time of the pulse. Another drawback is that practically, it is not possible to adjust the pulse width. The highest width of high voltage pulse is around 7 ns [35] and the highest achieved frequency is 200 kHz [28]. Pockels cells are characterized by higher contrast ratio and shorter duration of rise/fall time in comparison with the acousto-optic modulators. However, the high voltage pulses with the short rise/fall duration can cause piezoelectrical ringing that induces acoustic waves in Pockels cell crystal. If acoustic waves are not suppressed, they are reflected back inside of the Pockels cell crystal. This phenomenon introduces the elasto-optic effect, which can cause the reduction of Pockels cells contrast ratio [11,36–39]. This effect has to be investigated in order to find the ways how to reduce the impact of the piezoelectric ringing on the contrast ratio. The beta barium borate (BBO) crystals are usually used for the implementation of Pockels cells. There are many works dedicated to the analysis of effects in BBO crystals, e.g., [11,40–43]. However, the piezoelectric ringing phenomenon in Pockels cells with BBO crystals was mentioned in merely one research [11] and there are none dedicated to the analysis of high voltage pulse frequency impact on piezoelectric ringing phenomenon. The investigation results of piezoelectric ringing of the Pockels cells with BBO crystals that operate in high voltage pulse frequency range from 10 kHz up to 1 MHz are presented. The method of the piezoelectric ringing investigation based on the spectrum analysis of difference of real and simulated optical signals is proposed. The investigations are performed for the crystals with 3 × 3 × 25 mm, 4 × 4 × 25 mm and 4 × 4 × 20 mm dimensions.

2. Investigation Procedure Pockels cells based on the BBO crystals were investigated experimentally. A block diagram of simplified single pass contrast ratio measurement setup is given in Figure1. A 1064 nm wavelength diode-pumper solid state (DPSS) laser with 1.5 mm beam diameter and 2 W power was used in this setup. Half-wave plate with antireflective (AR) coating at 1064 nm was used to alter beam polarization of the DPSS laser. Glan-Taylor polarizing prism was selected for its high extinction ratio, which is higher than 100,000:1 [44]. This prism is used in order to let through only the p-polarized beam. The optical attenuator in combination with half-wave plate was employed in order to adjust the power and to avoid the saturation of photodetector. Laser beam from the first Glan-Taylor polarizing prism travels through the Pockels cell and after that to the second Glan-Taylor prism. The light intensity was detected via a high-speed Si photodetector (DET10A/M, Thorlabs, Newton, NJ, United States, 2013). It must be noted that the phases of the first and the second Glan-Taylor polarizing prisms were mutually turned by 90◦, i.e., the second Glan-Taylor polarizing prism was used as an analyzer. Crystals 2019, 9, 49 3 of 10 Crystals 2019, 9 FOR PEER REVIEW 3

PULSE PERSONAL GENERATOR COMPUTER DPSS LASER

HIGH VOLTAGE HIGH VOLTAGE OSCILLOSCOPE POWER SUPPLY DRIVER

MIRROR HALF-WAVE GLAN-TAYLOR POCKELS GLAN-TAYLOR PHOTO PLATE PRISM CELL PRISM DETECTOR FigureFigure 1. 1. SingleSingle pass pass contrast contrast ratio ratio measurement measurement setup, setup, where where DPSS DPSS - - diode-pumper diode-pumper solid solid state. state.

BipolarBipolar high voltagevoltage powerpower supply supply (PS2-60-1.4, (PS2-60-1.4, Eksma Eksma Optics, Optics, Vilnius, Vilnius, Lithuania, Lithuania, 2015) 2015) was used.was used.It provided It provided positive positive and negative and negative output output voltage voltage 1 kV to 1 1.4kV kV.to 1.4 A potentialkV. A potential difference difference of 2.8 kV of was2.8 kVpossible was withpossible this device.with this Voltage device. was Voltage supplied was to the supplied bipolar highto the voltage bipolar driver high (DPD-1000-2.9-Al, voltage driver (DPD-1000-2.9-Al,Eksma Optics, Vilnius, Eksma Lithuania, Optics, Vilnius, 2015), whichLithuania, produced 2015), which high voltage produced pulses high and voltage operated pulses on and the operatedMOSFET on connected the MOSFET in series connected principle. in series Pulse princi widthple. adjustment Pulse width from adjustment 100 ns up tofrom 5 µ s100 was ns achieved up to 5 μwiths was this achieved driver with model. this Highdriver voltage model. pulse High rise/fallvoltage pulse edge rise/fall duration edge from duration 8 ns down from to 8 ns 4 nsdown was toachievable 4 ns was achievable at pulse frequencies at pulse frequencies up to 1 MHz. up Frequencyto 1 MHz. Frequency and duration and of duration the high of voltage the high pulses voltage was pulsesassigned was via assigned pulse generator via pulse (9530, generator Quantum (9530, composers, Quantum Bozeman,composers, MT, Bozeman, United States,MT, United 2014). States, 2014).High voltage pulses from bipolar high voltage driver were supplied to the Pockels cell. When the voltageHigh was voltage applied pulses to the from Pockels bipolar cell, high the output voltage signal driver of were the photodetector supplied to the became Pockels high. cell. When When no thevoltage voltage was was applied applied to the to Pockelsthe Pockels cell, photodetectorcell, the output output signal signal of the was photodetector low. Oscilloscope became sampled high. Whenthe voltage no voltage on the was output applied of the to photodetector the Pockels cell, and ph capturedotodetector the oscillograms. output signal The was intensity low. Oscilloscope of the laser sampledbeam that the was voltage recorded on the by output the photodetector of the photodet canector be calculated and captured by [45 the]: oscillograms. The intensity of the laser beam that was recorded by the photodetector can be calculated by [45]: 2 Γ I = I0 sin , (1) 𝐼=𝐼sin2 , (1) where I is the intensity of incident light and Γ is phase retardation, which is calculated using equation: where I00 is the intensity of incident light and Γ is phase retardation, which is calculated using equation: 2 2πLn0rijV Γ = , (2) 2π𝐿𝑛λ r𝑉 𝛤= d , (2) λ𝑑 where L is the crystal length, n0 is the refractive index, rij is the electro-optic coefﬁcient, V is the voltage whereapplied, L λis isthe the crystal wavelength length, of n theo is laser the refractive beam and index,d is the r crystalij is the thickness. electro-optic coefficient, V is the voltageContrast applied, ratio λ is was the calculatedwavelength using of the the laser intensity beam valueand d ofis the optical crystal signal, thickness. which was captured by the oscilograms,Contrast ratio applying was calculated equation using [46]: the intensity value of optical signal, which was captured by the oscilograms, applying equation [46]: 1 1 1 1 = , (3) CR =Imax 𝐶𝑅 𝐼 , (3) Imin 𝐼 where:where: 1 Z tpw = ( ) Imax Ud t dt , (4) t0 − tpw1 t 𝐼 = 0 𝑈(𝑡)𝑑𝑡 , (4) 𝑡 −𝑡 1 Ztp Imin = Ud(t)dt , (5) tp − tpw t 1 pw 𝐼 = 𝑈(𝑡)𝑑𝑡 𝑡 −𝑡 , (5)

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where Ud(t) is the voltage of photodetector, t0 is moment at which the optical pulse starts, tpw is the Crystals 2019, 9 FOR PEER REVIEW 4 width of optical pulse and tp is the period of the pulses. Imax and Imin are the average of maximum and minimum intensity of the optical signal in the input of the photodetector. where Ud(t) is the voltage of photodetector, t0 is moment at which the optical pulse starts, tpw is the The contrast ratio of Pockels cells was measured in the high voltage pulse frequency range 10 kHz width of optical pulse and tp is the period of the pulses. Imax and Imin are the average of maximum and up to 1 MHz. Oscilloscope data were sampled and the frequency was changed with the step of 3 kHz minimum intensity of the optical signal in the input of the photodetector. automaticallyThe contrast via softwareratio of Pockels developed cells bywas the measured authors. in the high voltage pulse frequency range 10 kHzMeasurements up to 1 MHz. Oscilloscope were performed data were for sampled the the and BBO the frequency Pockels cellwas crystalschanged with with the the step following of 3 dimensions:kHz automatically via software developed by the authors. Measurements were performed for the the BBO Pockels cell crystals with the following • 3 mm × 3 mm × 25 mm dimensions: • × × 4 mm •4 mm3 mm25 × 3 mm mm × 25 mm • 4 mm ×•4 mm4 mm× 20 × 4 mm mm × 25 mm • 4 mm × 4 mm × 20 mm 3. Investigation Results 3. InvestigationPhotodetector Results signal oscilograms for Pockels cells with BBO crystals of 3 × 3 × 25 mm, 4 × 4 ×Photodetector25 mm and signal4 × 4 ×oscilograms20 mm dimensions for Pockels were cells capturedwith BBO atcrystals high voltageof 3 × 3 × pulses 25 mm, in 4 ranges × 4 × 25 from 10mm kHz and to 14 MHz× 4 × 20 with mm 3 dimensions kHz steps. Photodetectorwere captured signalat high for voltage Pockels pulses cell in with ranges3 mm from× 310 mm kHz× to25 1 mm BBOMHz crystal with 3 is kHz presented steps. Photodetector in Figure2. Additionally, signal for Pockels the simulated cell with square3 mm × wave 3 mm signal × 25 mm of photodetector BBO crystal is presentedis presented in Figure in Figure2. The 2. simulation Additionally, was performedthe simulated using square LabVIEW wave software signal of (National photodetector Instruments, is Austin,presented TX, in United Figure States, 2. The 2016) simulation the model was usedperformed for simulation using LabVIEW did not software take into (National account the piezoelectricInstruments, ringing Austin, phenomenon TX, United States, of the Pockels2016) the cell model crystal. used for simulation did not take into account the piezoelectric ringing phenomenon of the Pockels cell crystal.

(a) (b)

Figure 2. Cont.

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(e) (f)

× × FigureFigure 2. 2.Real Real and and simulated simulated photodetector photodetector signal fo forr Pockels Pockels cell cell with with 3 3mm mm × 3 mm3 mm × 25 mm25 mm beta beta bariumbarium borate borate (BBO) (BBO) crystal crystal at at frequencies frequencies of of high high voltage voltage pulses: pulses: ( a()a 150) 150 kHz; kHz; ( b(b) 180) 180 kHz; kHz; ( c()c 205) 205 kHz; (d)kHz; 445 kHz;(d) 445 (e kHz;) 600 ( kHz;e) 600 ( fkHz;) 750 (f kHz.) 750 kHz.

TheThe photodetector photodetector signal signal was was close close to to the the simulated simulated square square wave wave signal, signal, just just at at the the 180 180 kHz kHz high voltagehigh voltage pulse frequencypulse frequency (Figure (Figure2b). This 2b). showedThis show thated that the the piezoelectric piezoelectric ringing ringing did did not not occur occur at at the 180the kHz 180 frequency. kHz frequency. However, However, in all in other all other oscillograms oscillograms that that are presentedare presented in Figure in Figure2, optical 2, optical signal wassignal distorted was distorted as compared as compared to the simulated to the squaresimulated wave square signal. wave Distortion signal. ofDistortion the photodetector of the signalphotodetector was the consequencesignal was the of consequence the piezoelectric of the ringingpiezoelectric phenomenon. ringing phenomenon. Oscilograms Oscilograms presented in Figurepresented2a,c–f in show Figures that 2a,c–f the period show that of oscillations the period of caused oscillations by the caused piezoelectric by the piezoelectric ringing phenomenon ringing dependedphenomenon on the depended high voltage on the pulsehigh voltage frequency. pulse Thefrequency. highest The distortion highest distortion of the optical of the signaloptical was recordedsignal was at the recorded 600 kHz at highthe 600 voltage kHz high pulse voltage frequency pulse (Figure frequency2e). This(Figure happens 2e). This because happens the because frequency ofthe frequency piezoelectric of the ringing piezoelectric coincided ringing with coincided the frequency with the of frequency high voltage of high pulses voltage in this pulses situation in this and, therefore,situation some and, therefore, resonance some occurred. resonance occurred. The signals caused by the piezoelectric ringing oscillations at 150 kHz, 205 kHz, 445 kHz, 600 The signals caused by the piezoelectric ringing oscillations at 150 kHz, 205 kHz, 445 kHz, 600 kHz kHz and 750 kHz frequencies were analyzed using discrete Fourier transform (DFT): and 750 kHz frequencies were analyzed using discrete Fourier transform (DFT): 𝑦(𝑓)=𝑥 𝑒/ N−1 , (4) y( f ) = x e−j2πkn/N (6) where n = 0,1,2, …, N-1, x is input sequence, N ∑is then number of elements of x and y is the transform n=0 result. where Spectrumsn = 0, 1, 2,of... real, N and−1, simulatedx is input photodetector sequence, N signalsis the numberat the 180 of elementskHz high ofvoltagex and pulsey is the transformfrequency result. obtained using DFT are presented in Figure 3a. The difference of real and simulated photodetectorSpectrums ofsignal real andspectrums simulated was photodetector calculated to signalsevaluate at the the spectrum 180 kHzhigh inequality. voltage The pulse result frequency of obtainedcalculation using is presented DFT are presentedin Figure 3b. in This Figure showed3a. The that difference spectrums of of real real andand simulatedsimulated signals photodetector were signalclose spectrumsto each other. was This calculated fact proves to that evaluate no piezoelectric the spectrum ringing inequality. oscillations The occurred result at of frequencies calculation is presentedup to 11 inMHz Figure if the3b. high This voltage showed pulses that with spectrums 180 kHz of were real used and simulatedfor the Pockels signals cell werecontrol. close to each other. This fact proves that no piezoelectric ringing oscillations occurred at frequencies up to 11 MHz if the high voltage pulses with 180 kHz were used for the Pockels cell control. The differences of real and simulated photodetector signal spectrums calculated for the signals obtained at the 150 kHz, 180 kHz, 205 kHz, 445 kHz and 750 kHz high voltage pulse frequencies are presented in Figure4. The results for 600 kHz frequency are not displayed, since the subtraction result was very high, and therefore, the imaging of the graph would be worsened.

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(a) (b)

Figure 3. (a) Spectrums of real and simulated photodetector signals; (b) result of spectrum subtraction. High voltage pulse frequency at 180 kHz.

The differences of real and simulated photodetector signal spectrums calculated for the signals (a) (b) obtained at the 150 kHz, 180 kHz, 205 kHz, 445 kHz and 750 kHz high voltage pulse frequencies are presentedFigure 3. in(a )Figure Spectrums 4. The of real results and simulatedfor 600 kHz photodetector frequency signals;are not (b displayed,) result of spectrum since the subtraction. subtraction Figure 3. (a) Spectrums of real and simulated photodetector signals; (b) result of spectrum resultHigh was voltage very pulse high, frequency and therefore, at 180 the kHz. imaging of the graph would be worsened. subtraction. High voltage pulse frequency at 180 kHz.

The differences of real and simulated photodetector signal spectrums calculated for the signals obtained at the 150 kHz, 180 kHz, 205 kHz, 445 kHz and 750 kHz high voltage pulse frequencies are presented in Figure 4. The results for 600 kHz frequency are not displayed, since the subtraction result was very high, and therefore, the imaging of the graph would be worsened.

FigureFigure 4. Real4. Real and and simulated simulated signals signalsspectrums spectrums subtractionsubtraction at at 150 150 kHz, kHz, 180 180 kHz, kHz, 205 205 kHz, kHz, 445 445 kHz kHz andand 750 750 kHz kHz high high voltage voltage pulse pulse frequencies. frequencies.

TheThe highest highest amplitudes amplitudes ofof spectrumspectrum componentscomponents obtained for for 150 150 kHz kHz high high voltage voltage pulse pulse frequencyfrequency were were observed observed at at 150 150 kHzkHz andand 450450 kHz;kHz; for for 205 205 kHz kHz at at 205 205 kHz kHz with with a alot lot of oflower lower amplitudeamplitude components, components, for for 445 445 kHz kHz at at 445 445 kHzkHz andand 890890 kHz, and for for 750 750 kHz kHz at at 750 750 kHz. kHz. TheThe piezoelectric piezoelectric ringing ringing in thein Pockelsthe Pockels cell withcell with BBO withBBO dimensionswith dimensions of 3 × of3 ×3 25× 3 mm × 25 occurred mm if theoccurredFigure spectrum 4. if Realthe components spectrumand simulated components of thesignals high spectrums of voltage the high pulsessubtraction voltage were pulses at near 150 were thekHz, resonant near180 kHz, the frequencies resonant205 kHz, frequencies445 of kHz 445kHz, 600 kHzand 750 or 750kHz kHz. high voltag Therefore,e pulse the frequencies. acoustic wave suppressors designed for these frequencies had to be applied. TheThe highest dependences amplitudes of Pockels of spectrum cell BBO componen crystals contrastts obtained ratio for on 150 applied kHz high high-voltage voltage pulse pulse frequencyfrequency were was obtainedobserved byat 150 processing kHz and the 450 captured kHz; for oscilograms 205 kHz at data205 kHz using with the a contrast lot of lower ratio amplitude components, for 445 kHz at 445 kHz and 890 kHz, and for 750 kHz at 750 kHz. The piezoelectric ringing in the Pockels cell with BBO with dimensions of 3 × 3 × 25 mm occurred if the spectrum components of the high voltage pulses were near the resonant frequencies

CrystalsCrystals 20192019,, 99 FORFOR PEERPEER REVIEWREVIEW 77 ofof 445kHz,445kHz, 600600 kHzkHz oror 750750 kHz.kHz. Therefore,Therefore, thethe acousticacoustic wavewave suppressorssuppressors designeddesigned forfor thesethese frequenciesCrystalsfrequencies2019, 9 ,hadhad 49 toto bebe applied.applied. 7 of 10 TheThe dependencesdependences ofof PockelsPockels cellcell BBOBBO crystalscrystals contrastcontrast ratioratio onon appappliedlied high-voltagehigh-voltage pulsepulse frequencyfrequency waswas obtainedobtained byby processingprocessing thethe capturcaptureded oscilogramsoscilograms datadata usingusing thethe contrastcontrast ratioratio EquationEquation (3).(3). TheThe obtainedobtained dependencesdependencesdependences ofofof 3 33 × ×× 33 ××× 2525 and and 44 ××× 44 ×× 252525 mmmm mm PockelsPockels cellcell BBO BBO crystalscrystals contrastcontrast on on applied applied high-voltagehigh-voltage pulsepulse frequencyfrequency areare presentedpresented ininin FigureFigureFigure5 5..5.

FigureFigure 5.5. TheThe dependencesdependences ofof PockelsPockels cellcell BBOBBO crystalscrystals contrastcontrast ratioratio onon appliedapplied high-voltagehigh-voltage pulsepulse frequencyfrequency for for crystalscrystals withwith variousvarious dimensions:dimensions: 1—41–1– 44 ××× 444 ××× 252525 mm;mm; mm; 2–2– 2—3 33 ×× 33× ××3 2525× mm.mm.25 mm. × × TheThe obtainedobtained resultsresults forfor PockelsPockelsPockels cellcellcell withwithwith 333 ×× 33 ×× 2525 mmmm BBO BBO crystal crystal show show thatthat piezoelectricpiezoelectric ringingringing occurredoccurred and,and, and, becausebecause because ofof of this,this, this, thethe the contrastcontrast contrast ratioratio ratio decreaseddecreased decreased atat at150150 150 kHz,kHz, kHz, 205205 205 kHzkHz kHz, 445445 kHz, 445kHz, kHz, 600600 kHz600kHz kHz andand 750 and750 kHzkHz 750 high kHzhigh voltage highvoltage voltage pulsepulse pulsefrequencies.frequencies. frequencies. SpSpectrumectrum Spectrum analysisanalysis analysis showsshows thatthat shows piezoelectricpiezoelectric that piezoelectric ringingringing atringingat thethe 150150 at kHzkHz the 150 andand kHz 205205 kHz andkHz 205frequenciesfrequencies kHz frequencies waswas causedcaused was byby caused thethe thirdthird by theharmonicharmonic third harmonic ofof highhigh voltagevoltage of high pulses.pulses. voltage ItIt pulses.isis seenseen It(Figure(Figure is seen 5)5) (Figure thatthat resonantresonant5) that resonant frequenciesfrequencies frequencies depedependednded depended onon thethe on sizesize the ofof size thethe of BBOBBO the BBOcrystalcrystal crystal aperture.aperture. aperture. TheThe Theresonancesresonances resonances ofof smallersmaller of smaller crystalcrystal crystal occurredoccurred occurred atat higherhigher at higher frfrequencies.equencies. frequencies. WhenWhen When apertureaperture aperture ofof crystalcrystal of crystal changeschanges changes fromfrom 3from3 toto 44 3mm,mm, to 4 thethe mm, lowestlowest the lowestresonanceresonance resonance frequencyfrequency frequency decreaseddecreased decreased byby 3636 kHzkHz by atat 36 thethe kHz 114114 at kHzkHz the frequency,frequency, 114 kHz frequency, whilewhile forfor whilethethe highesthighest for the resonanceresonance highest resonance frequencyfrequency frequency itit decreasedecrease itdd decreased byby 195195 kHzkHz by atat 195 thethe kHz 550550 at kHzkHz the frequency.frequency. 550 kHz frequency. ItIt isis seenseen seen thatthat that withwith with thethe the decreasingdecreasing decreasing ofof of BBOBBO BBO PockPock Pockelselsels cellcell cell crystalcrystal crystal lengthlength length fromfrom from 2525 mmmm 25 mm toto 2020 to mm,mm, 20 mm, thethe resonancetheresonance resonance frequencyfrequency frequency shiftedshifted shifted slightlyslightly slightly (Figure(Figure (Figure 6).6). 6).

FigureFigure 6.6. TheThe dependencesdependences ofof PockelsPockels cellscells BBOBBO crystalcrystal contrastcontrast ratioratio onon appliedapplied high-voltagehigh-voltage pulsepulse frequencyfrequency for for crystalscrystals withwith variousvarious dimensions:dimensions: 1—41–1– 44 ××× 444 ××× 202020 mm;mm; mm; 22 2—4–– 44 ×× ×44 ××4 2525× 25mm.mm. mm.

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4. Discussion The novel method of piezoelectric ringing analysis of Pockels cells with BBO crystals, which is based on the subtraction of real and simulated photodetector signals spectrums, allows us to observe spectrum components caused by the piezoelectric ringing phenomenon. The obtained results for Pockels cell with 3 × 3 × 25 mm BBO crystal show that piezoelectric ringing occurred and, because of this, the contrast ratio decreased at 150 kHz, 205 kHz, 445 kHz, 600 kHz and 750 kHz high voltage pulse frequencies. Spectrum analysis shows that piezoelectric ringing at the 150 kHz and 205 kHz frequencies was caused by the third harmonic of high voltage pulses. The frequencies at which the piezoelectric ringing occurred depend on the size of the BBO crystal aperture. The piezoelectric ringing of smaller crystal occurred at higher frequencies. When aperture of crystal changed from 3 to 4 mm, the lowest frequency decreased by 36 kHz at 114 kHz frequency, while for the highest resonance frequency it decreased by 195 kHz at the 550 kHz frequency. The highest distortion of the optical signal was recorded at the 600 kHz high voltage pulse frequency. This happened because the frequency of the piezoelectric ringing coincided with the frequency of high voltage pulses in this situation, and therefore, some resonance occurred. The method for piezoelectric ringing suppression in Pockels cells with BBO crystals has to be developed to decrease the impact of this phenomenon on the contrast ratio of Pockels cell.

Author Contributions: G.S. and A.B. wrote the paper and analyzed the data. G.S. performed the experiments. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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